Operator-state correspondence is usual in $d\geq 2$. See for example Operator-state correspondence in QFT.
Is some kind of Operator-state correspondence in 1d CFT or more generally in quantum mechanics? Is such correspondence for SYK model?
In CFT essential step is radial quantisation. What is analogue in QM?
Indeed there is a version of the state operator correspondence that holds in 1d CFT/Quantum mechanics. Note that operator -> state map is trivially true in any general QFT in arbitrary dimensions. However, state -> operator map is the non-trivial bit that holds only in a CFT, and follows from scale invariance as depicted by the diagram in your question. In a 1-dimensional theory, due to lack of spatial dimensions, it is always possible to associate a state with an operator insertion. So my guess would be, that it is more generally true. However, a correspondence akin to the one in higher dimensions is discussed in https://arxiv.org/abs/1101.4254. You will find that in the case of 1-dimensional conformal theory, a operator insertion corresponds to a state in a double copy of theory.
